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The binding energies (BE) of molecules on the interstellar grains are crucial in the chemical evolution of the interstellar medium (ISM). Both temperature programmed desorption (TPD) laboratory experiments and quantum chemistry computations have often provided, so far, only single values of the BE for each molecule. This is a severe limitation, as the ices enveloping the grain mantles are structurally amorphous, giving rise to a manifold of possible adsorption sites, each with different BEs. However, the ice amorphous nature prevents the knowledge of structural details, hindering the development of a common accepted atomistic icy model. In this work, we propose a computational framework that closely mimics the formation of the interstellar grain mantle through a water by water accretion. On that grain, an unbiased random (but well reproducible) positioning of the studied molecule is then carried out. Here we present the test case of NH$_3$, an ubiquitous species in the molecular ISM. We provide the BE distribution computed by a hierarchy approach, using the semiempirical xTB-GFN2 as low-level method to describe the whole icy cluster combined with the B97D3 DFT functional as high-level method on the local zone of the NH$_3$ interaction. The final ZPE corrected BE is computed at ONIOM(DLPNO-CCSD(T)//B97D3:xTB-GFN2) level, ensuring the best cost/accuracy ratio. The main peak of the predicted NH$_3$ BE distribution is in agreement with experimental TPD and literature computed data. A second broad peak at very low BE values is also present, never detected before. It may provide the solution to a long-standing puzzle about the presence of gaseous NH$_3$ observed also in cold ISM objects.